Solve the Function Operation f(x)=x^2+3x+4 , g(x)=x-1 , fog

$f (x) = x^{2} + 3 x + 4$ , $g (x) = x - 1$ , $(f \circ g)$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x - 1)$ by substituting in the value of $g$ into $f$ .

$f (x - 1) = {(x - 1)}^{2} + 3 (x - 1) + 4$

Step 3

Simplify each term.

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$f (x - 1) = (x - 1) (x - 1) + 3 (x - 1) + 4$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$f (x - 1) = x (x - 1) - 1 (x - 1) + 3 (x - 1) + 4$

Apply the distributive property.

$f (x - 1) = x \cdot x + x \cdot - 1 - 1 (x - 1) + 3 (x - 1) + 4$

Apply the distributive property.

$f (x - 1) = x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x - 1) = x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Move $- 1$ to the left of $x$ .

$f (x - 1) = x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Rewrite $- 1 x$ as $- x$ .

$f (x - 1) = x^{2} - x - 1 x - 1 \cdot - 1 + 3 (x - 1) + 4$

Rewrite $- 1 x$ as $- x$ .

$f (x - 1) = x^{2} - x - x - 1 \cdot - 1 + 3 (x - 1) + 4$

Multiply $- 1$ by $- 1$ .

$f (x - 1) = x^{2} - x - x + 1 + 3 (x - 1) + 4$

Subtract $x$ from $- x$ .

$f (x - 1) = x^{2} - 2 x + 1 + 3 (x - 1) + 4$

Apply the distributive property.

$f (x - 1) = x^{2} - 2 x + 1 + 3 x + 3 \cdot - 1 + 4$

Multiply $3$ by $- 1$ .

$f (x - 1) = x^{2} - 2 x + 1 + 3 x - 3 + 4$

Step 4

Simplify by adding terms.

Add $- 2 x$ and $3 x$ .

$f (x - 1) = x^{2} + x + 1 - 3 + 4$

Simplify by adding and subtracting.

Subtract $3$ from $1$ .

$f (x - 1) = x^{2} + x - 2 + 4$

Add $- 2$ and $4$ .

$f (x - 1) = x^{2} + x + 2$